Judging whether a finitely presented group is a 3-manifold group? Given a finitely presented group $G$, how many necessary conditions do people know for $G$ to be isomorphic to the fundamental group of some closed connected 3-manifold?  (e.g. residually finite)
 A: since Henry started the shameless self-promotion, let me also do so...
Given any group $\pi$ one can study the corresponding Alexander polynomial $\Delta_\pi$ which lies
in the group ring of $H:=H_1(\pi;\Bbb{Z})/\mbox{torsion}$. 
If $\pi$ is the fundamental group of a closed 3-manifold, then the 
Alexander polynomial $\Delta_{\pi}$ is symmetric and the one-variable specializations 
have even degree.
(see F, Kim, Kitayama: Poincaré duality and degrees of twisted Alexander polynomials)
The symmetry holds also if $\pi$ is a 3-dimensional Poincare duality group, but I am not sure whether the degree condition holds in that case.
The advantage is that this condition can be checked easily, and by checking it for finite index subgroups one gets even more necessary conditions. I would guess that in practice this is a very effective way for weeding out non 3-manifold groups.
At least it allowed me to make the right bet on Ryan's example...
A: As a demonstration of difficulty, a counter-question: Is this a 3-manifold fundamental group, and if so, which one? 
$$\langle a, b  | a^2b^{-1}a^{-2}ba^{-1}ba^{-2}b^{-1}a^2b, a^{-1}b^{-1}a^2ba^{-4}ba^2b^{-1}a^{-1}b^{-1}a^2ba^{-2}ba^2b^{-1} \rangle$$
A: Apologies for the shameless self-promotion, but as you ask for necessary conditions, you seem to want a list of theorems of the form 'If G is a 3-manifold group then G has property P'.
Aschenbrenner, Friedl and I have given what I think is a pretty exhaustive list in our survey paper here.
